# On superconvergence of Runge-Kutta convolution quadrature for the wave equation

## TL;DR

This paper analyzes the convergence properties of Runge-Kutta convolution quadrature methods for wave equation scattering problems, demonstrating improved convergence rates and providing new estimates on Dirichlet-to-Impedance maps.

## Contribution

It introduces a novel estimate on the Dirichlet-to-Impedance map that enhances understanding of superconvergence in Runge-Kutta convolution quadrature for wave equations.

## Key findings

- Higher convergence rate for the derivative-based Runge-Kutta method.
- Numerical results confirm the sharpness of the theoretical analysis.
- New estimate reduces frequency dependence in Helmholtz problem maps.

## Abstract

The semidiscretization of a sound soft scattering problem modelled by the wave equation is analyzed. The spatial treatment is done by integral equation methods. Two temporal discretizations based on Runge-Kutta convolution quadrature are compared: one relies on the incoming wave as input data and the other one is based on its temporal derivative. The convergence rate of the latter is shown to be higher than previously established in the literature. Numerical results indicate sharpness of the analysis. The proof hinges on a novel estimate on the Dirichlet-to-Impedance map for certain Helmholtz problems. Namely, the frequency dependence can be lowered by one power of $\abs{s}$(up to a logarithmic term for polygonal domains) compared to the Dirichlet-to-Neumann map.